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Cayley diagram

C22×A5 #6829

order240
isoclinismisoclinic class of A5
automorphismAut(G) = D3×S5 (order 720) · Inn(G) = A5 · Out(G) = C2×D3
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C22

Aliases

direct productC22×A5

Subgroups

subgroupG/H
center, Fitting, hypercenterC22A5
commutator, hypocenterA5C22
FrattiniC1G
LCSG ▷ A5
UCSC1 ◁ C22

71 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 9
transitive degreeμt(G) = 20

A minimal faithful permutation representation — 2 generators on 9 points:

(1,2)(3,4)(5,6,9);(3,4)(5,6,7,8,9)

Conjugacy classes & character table

20 conjugacy classes — view the full character table.

Relationships

derived subgroup ofA4×A5, A4×S5, C2×A4×A5, …
socle ofC2×A5⋊C4, C2×C4×A5, C22×S5, …
maximal subgroup ofC22⋊S5, C2×A5⋊C4, C2×C4×A5, …
maximal normal subgroup ofC22⋊S5, C2×A5⋊C4, C2×C4×A5, …
maximal quotient ofC2×C4×A5, C2×#6732, D4×A5, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 190);